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Gravitational quadrupolar coupling to equivalence principle test masses: the general case

Lockerbie, N.A. (2002) Gravitational quadrupolar coupling to equivalence principle test masses: the general case. Classical and Quantum Gravity, 19 (8). pp. 2063-2077. ISSN 0264-9381

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This paper discusses the significance of the quadrupolar gravitational force in the context of test masses destined for use in equivalence principle (EP) experiments, such as STEP and MICROSCOPE. The relationship between quadrupolar gravity and rotational inertia for an arbitrary body is analysed, and the special, gravitational, role of a body's principal axes of inertia is revealed. From these considerations the gravitational quadrupolar force acting on a cylindrically symmetrical body, due to a point-like attracting source mass, is derived in terms of the body's mass quadrupole tensor. The result is shown to be in agreement with that obtained from MacCullagh's formula (as the starting point). The theory is then extended to cover the case of a completely arbitrary solid body, and a compact formulation for the quadrupolar force on such a body is derived. A numerical example of a dumb-bell's attraction to a local point-like gravitational source is analysed using this theory. Close agreement is found between the resulting quadrupolar force on the body and the difference between the net and the monopolar forces acting on it, underscoring the utility of the approach. A dynamical technique for experimentally obtaining the mass quadrupole tensors of EP test masses is discussed, and a means of validating the results is noted.